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Page "Burnside's problem" ¶ 27
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Let and M
Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.
Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.
Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.
Let M be a smooth manifold and let f ∈ C < sup >∞</ sup >( M ) be a smooth function.
Let r be a non zero real number and let the r < sup > th </ sup > power mean ( M < sup > r </ sup > ) of a series of real variables ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ... ) be defined as
Let M be a ( pseudo -) Riemannian manifold, which may be taken as the spacetime of general relativity.
Let M and N be ( left or right ) modules over the same ring, and let f: M → N be a module homomorphism.
Let M be an n × n Hermitian matrix.
Let P < sup >− 1 </ sup > DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues.
Let ( M, g ) be a Riemannian manifold and ƒ: M < sup > m </ sup > → R < sup > n </ sup > a short C < sup >∞</ sup >- embedding ( or immersion ) into Euclidean space R < sup > n </ sup >, where nm + 1.
Let M and N be smooth manifolds and be a smooth map.

Let and be
Let the open enemy to it be regarded as a Pandora with her box opened ; ;
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
`` Let him be now ''!!
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let T be a linear operator on the finite-dimensional vector space V over the field F.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let Af be the null space of Af.
Let N be a linear operator on the vector space V.
Let T be a linear operator on the finite-dimensional vector space V over the field F.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let this be denoted by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let not your heart be troubled, neither let it be afraid ''.
The same God who called this world into being when He said: `` Let there be light ''!!
For those who put their trust in Him He still says every day again: `` Let there be light ''!!
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let her out, let her out -- that would be the solution, wouldn't it??

Let and intersection
Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.
Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:
# Let B < sub > 1 </ sub >, B < sub > 2 </ sub > be base elements and let I be their intersection.
Let Q and R be the points of intersection of these two circles.
Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.
Let there be a right angle ABC, r a line parallel to BC passing by A and s a line parallel to AB passing by C. Let D be the point of intersection of lines r and s ( Note that it has not been proven that D lies on the circle )
Let O be the point of intersection of the diagonals AC and BD.
Let the S be a sphere with center O, P a plane which intersects S. Draw perpendicular to P and meeting P at E., Let A and B be any two points in the intersection.
Let M be an n-dimensional manifold, and let k be an integer such that 0 ≤ k ≤ n. A k-dimensional embedded submanifold of M is a subset S ⊂ M such that for every point p ∈ S there exists a chart ( U ⊂ M, φ: U → R < sup > n </ sup >) containing p such that φ ( S ∩ U ) is the intersection of a k-dimensional plane with φ ( U ).
Another construction proceeds as follows: Let a line L and a point O not on L be given, and let K be the line through O parallel to L. Let P be a variable point on L. Let Q be the projection of P onto K, on other words Q is the intersection of K with the line through P perpendicular to K. Similarly let R be the projection of Q onto OP.
Let C be the point of intersection of J with OA.
Denote the intersection of AC and BD by M. Drop the perpendicular from M to the line BC, calling the intersection E. Let F be the intersection of the line EM and the edge AD.
Consider two rays emanating from an external homothetic center E. Let the antihomologous pairs of intersection points of these rays with the two given circles be denoted as P and Q, and S and T, respectively.
Let point P be the intersection of lines AC and BD.
Let us consider an axis Δ, and planes belonging to the zone of this axis, i. e. Δ is in all these planes, the intersection of all the planes is Δ.
Let be the point of intersection, then the angle formed by the lines at is.

1.125 seconds.